Strict Positive Realness As a Mapping

There are some different statements for necessary and sufficient conditions of strict positive realness in the control literatures. These statements are usually extracted by using state space approaches. In this paper, unlike the other works the frequency domain tools is used to prove the necessary and sufficient conditions for test strict positive realness property of real rational transfer function. The first main advantages of this approach is develop a new sufficient condition for strict positive realness and the second is prove necessary and sufficient conditions for strict positive realness property of real rational transfer function directly by using original definition in the frequency domain and without any restriction on the relative degree of transfer function by default. In this paper, the original definition of strict positive realness in the frequency domain is interpreted as a mapping from S-plane to G(s)-plane. Then the maximum modulus principle and the Taylor expansion which are the fundamental tools in the complex analysis is used to extract a new sufficiency condition for strictly positive real (SPR) transfer functions. This geometry interpretation implies a new more accurate definition for SPR transfer functions. Finally more completed and simplified version of necessary and sufficient conditions for strict positive realness is proved based on the complex analysis